A major problem in single photon emission computed tomography (SPECT) is the failure of the reconstruction algorithm to provide accurate quantitative information about the uptake of radionuclide in various organs. With numerous new imaging agents and applications becoming available, crude images of the radionuclide distribution within the body are no longer sufficient. Physiological studies require accurate esti- mates of radionuclide uptake in specific organs. Unfortunately, there are significant physical limitations to the accuracy of quantitation in SPECT. Luckily, recent advances in computer technology raise the possibility of overcoming these limitations by new, mathematically sophisticated, reconstruction algorithms. One well-known problem is the trade-off between gamma-camera sensitivity and resolution, in which the requirement of high resolution leads to fewer counts and increased image noise. A promising approach to this problem is the use of converging-hole or astigmatic collimators which make more efficient use of the camera detector crystal. However, these collimators require reconstruction algorithms which are intrinsically 3-dimensional (rather than the usual 2-dimensional SPECT algorithms which reconstruct each slice independently). Consequently, the development of algorithms for converging-hole collimators is currently an active field of research. Another significant problem in SPECT imaging is the attenuation and scatter of the radiation within the patient. Because the attenuation and scatter depend upon source location within the body and the intervening tissue, accurate compensation is difficult. However, information supplied by correlated CT or MRI images and by the energy resolution of the gamma camera may improve SPECT quantitation provided that reconstruction algorithms can be developed which utilize the new information. The goal of this project is the development of new algorithms addressing these problems. Algorithms for astigmatic collimators will be developed and tested using computer simulations because of the difficulty in obtaining astigmatic collimators with arbitrary focusing. Traditional methods are inadequate to describe the imaging properties of such collimators. A few years ago, we developed the generalized collimator transfer function (GCTF) which is ideally suited to deal with the non-stationary effects of convergent or astigmatic collimation. The GCTF will be used to simulate the SPECT projected images and as a tool in algorithm development. Image noise will be included in the projected images to test the noise immunity of any reconstruction algorithm. The algorithms which we intend to test are generalizations of the weighted filtered-backprojection reconstructions for conebeams developed by Feldcamp, et.al. and Peyrin, and the reconstruction algorithms for uncollimated PET of Defrise, et.al. Algorithms for attenuation and scatter correction will be based on my Ph.D. thesis research and will be tested using experimental phantom data.